A307842 Maximum number of nontrivial Latin subrectangles in a diagonal Latin square of order n.

0, 0, 0, 12, 12, 51, 151, 924

Offset

1,4

Comments

A Latin subrectangle is an m X k Latin rectangle of a Latin square of order n, 1 <= m <= n, 1 <= k <= n.

A nontrivial Latin subrectangle is an m X k Latin rectangle of a Latin square of order n, 1 < m < n, 1 < k < n.

Links

Table of n, a(n) for n=1..8.

E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian).

E. I. Vatutin, About the minimum and maximum number of nontrivial Latin subrectangles in a diagonal Latin squares of order 8 (in Russian).

Eduard Vatutin, Alexey Belyshev, Natalia Nikitina, and Maxim Manzuk, Evaluation of Efficiency of Using Simple Transformations When Searching for Orthogonal Diagonal Latin Squares of Order 10, High-Performance Computing Systems and Technologies in Sci. Res., Automation of Control and Production (HPCST 2020), Communications in Comp. and Inf. Sci. book series (CCIS, Vol. 1304) Springer (2020), 127-146.

Eduard I. Vatutin, Proving list (best known examples).

Index entries for sequences related to Latin squares and rectangles.

Example

For example, the square
  0 1 2 3 4 5 6
  4 2 6 5 0 1 3
  3 6 1 0 5 2 4
  6 3 5 4 1 0 2
  1 5 3 2 6 4 0
  5 0 4 6 2 3 1
  2 4 0 1 3 6 5
has nontrivial Latin subrectangle
  . . . . . . .
  . . 6 5 0 1 3
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . . . . . .
  . . 0 1 3 6 5
The total number of Latin subrectangles for this square is 2119 and the number of nontrivial Latin subrectangles is only 151.

Crossrefs

Cf. A307840, A307841.

Sequence in context: A303646 A298036 A119877 * A147833 A003877 A161196

Adjacent sequences: A307839 A307840 A307841 * A307843 A307844 A307845

Keyword

nonn,more,hard

Author

Eduard I. Vatutin, May 01 2019

Extensions

a(8) added by Eduard I. Vatutin, Oct 06 2020

Status

approved